^^ That's a common mistake. The focal length remains the same despite the crop factor. So, the image is enlarged by that factor but the focal length and how that affects the appearance remains the same. In other words, if you move back just enough to counter the crop factor, it would look the same as a 50mm on a full frame (but closer to the subject).
Imagine the circle of light that goes through the lens. Now, the image in the light doesn't change based on the sensor in your camera. Rather, full frame and crop sensors just capture a different portion of that light without changing how the actual image in the light appears. It's difficult to explain but basically the sensor size isn't changing the information in the light from the lens.
No, that is not correct. The focal length of the lens only determines one thing, and that's the field of view. However the perspective of the image is determined by your positioning relative to the objects you're shooting. As a result, if you need to step back with a crop frame camera to get the same objects in the frame you have changed the perspective and you are no longer capturing the same image. You may have captured the same objects at the boundaries of the image but the spatial relationships will all be altered.
Therefore, if you have the exact framing and composition with your eyeballs and you then look through your 50mm on a 1.6x crop you cannot take the image of what you see. Either your field of view will be narrower, meaning you've lost the edges of your frame, or you have to take a few steps back which means the perspective of your shot has changed. That's why a 30mm is closer to a normal lens then a 30mm on a crop frame... the field of view on the image is as close as you can get to your eyes and you don't have to change the perspective by moving to get the same angle.